* Prove and apply properties of parallelograms * Use properties of parallelograms to solve problems

1. Parallelograms & Rectangles
The student is able to (I can):
• Prove and apply properties of parallelograms.
• Use properties of parallelograms to solve problems.
• Prove and apply properties of special parallelograms.
• Use properties of special parallelograms to solve
problems.

2. parallelogram
Properties of
Parallelograms
A quadrilateral with two pairs of parallel
sides.
Therefore, if a quadrilateral is a
parallelogram, then it has two pairs of
parallel sides.
>>
>>
T I
ME
TI ME, TE IM

3. Properties of
Parallelograms
If a quadrilateral is a parallelogram, then
opposite sides are congruent.
If a quadrilateral is a parallelogram, then
opposite angles are congruent.
KI NG, GK IN≅ ≅
K
NG
I
>>
>>
K
NG
O
∠K ≅ ∠N, ∠O ≅ ∠G

4. Properties of
Parallelograms
If a quadrilateral is a parallelogram, then
consecutive angles are supplementary.
If a quadrilateral is a parallelogram, then
its diagonals bisect each other.
1 2
34
>>
>>
T U
NE
S
TS NS,ES US≅ ≅
∠ + ∠ = °
∠ + ∠ = °
∠ + ∠ = °
∠ + ∠ = °
m 1 m 2 180
m 2 m 3 180
m 3 m 4 180
m 4 m 1 180

5. Examples Find the value of the variable:
1. x =
2. x =
3. y =
5x + 3 2x + 15
(3x)º
(x + 84)º
yº

6. Examples Find the value of the variable:
1. x =
2. x =
3. y =
5x + 3 2x + 15
4
(3x)º
(x + 84)º
yº
5x + 3 = 2x + 15
3x = 12
3x = x + 84
2x = 84
42
3(42) = 126º
y = 180 — 126
54

7. rectangle A parallelogram with four right angles.
If a parallelogram is a rectangle, then its
diagonals are congruent (“checking for
square”).
F I
SH
≅FS IH

8. Because a rectangle is a parallelogram, it
also “inherits” all of the properties of a
parallelogram:
• Opposite sides parallel
• Opposite sides congruent
• Opposite angles congruent (actually allallallall
angles are congruent)
• Consecutive angles supplementary
• Diagonals bisect each other

9. Example Find each length.
1. LW
2. OL
3. OW
F O
WL
30
17

10. Example Find each length.
1. LW
LW = FO = 30
2. OL
OL = FW = 2(17) = 34
3. OW
∆OWL is a right triangle, so
OW = 16
F O
WL
30
17
+ =2 2 2
OW LW OL
+ =2
OW 900 1156
=2
OW 256
+ =2 2 2
OW 30 34

11. rhombus A parallelogram with four congruent sides.
(Plural is either rhombi or rhombuses.)
If a parallelogram is a rhombus, then its
diagonals are perpendicular.

12. If a parallelogram is a rhombus, then each
diagonal bisects a pair of opposite angles.
∠1 ≅ ∠2
∠3 ≅ ∠4
∠5 ≅ ∠6
∠7 ≅ ∠8
1 2 3
4
5
67
8
Since opposite angles are
also congruent:
∠1 ≅ ∠2 ≅ ∠5 ≅ ∠6
∠3 ≅ ∠4 ≅ ∠7 ≅ ∠8

13. Examples 1. What is the perimeter of a rhombus
whose side length is 7?
2. Find the value of x
3. Find the value of y
x
8
Perimeter = 40
(3y+11)º
(13y—9)º
10

14. Examples 1. What is the perimeter of a rhombus
whose side length is 7?
4(7) = 28
2. Find the value of x
The side = 10
Pyth. triple: 6, 8, 10
x = 6
3. Find the value of y
13y — 9 = 3y + 11
10y = 20
y = 2
x
8
Perimeter = 40
(3y+11)º
(13y—9)º
10

15. square A quadrilateral with four right angles and
four congruent sides.
Note: A square has all of the properties of
bothbothbothboth a rectangle andandandand a rhombus:
• Diagonals are congruent
• Diagonals are perpendicular
• Diagonals bisect opposite angles.